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<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="OrderClassGroup"></a>
OrderClassGroup
</h2>

Computes the class group.

<p><p>

<h3>
Syntax:
</h3>
<pre>L := OrderClassGroup( O [,b] [,"fast"] [,"Euler"] );</pre>

<p>

<table>
<tr>
<td>list</td>
<td><pre>  L  </pre></td>
<td></td>
</tr>
<tr>
<td>order</td>
<td><pre>  O  </pre></td>
<td></td>
</tr>
<tr>
<td>integer</td>
<td><pre>  b  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#OrderClassGroupCheck">OrderClassGroupCheck</a>, <a href="kashref.html#OrderClassGroupCyclicFactors">OrderClassGroupCyclicFactors</a>, <a href="kashref.html#IdealClassRep">IdealClassRep</a>, <a href="kashref.html#IdealIsPrincipal">IdealIsPrincipal</a>

<p>

<h3>
Description:
</h3>
The <tt> OrderClassGroup</tt> function computes the class group of
the algebraic number field which is generated by the order
O. It returns a list <tt> L</tt> whose first entry is the class
number. The second entry is a list which contains the
orders of the cyclic subgroups.

Notice that the class group is stored in the order O. To
get more information on the class group print out the order.

Several (optional) arguments can be passed to this function:
The first argument has always to be the order. As a second
argument a bound <tt> b</tt> for prime ideals to be considered
may be given. If it is omitted, <tt> OrderClassGroup</tt> uses
the Minkowski bound for the computation of the class group.
The Minkowski bound always guarantees correct
results. However, when the field discriminant is {\em
large}, the Minkowski bound requires very time consuming
computations.

Additionally, the following options can be
passed to speed up the computation:
<tt> "fast"</tt> indicates a fast computation without checking
the results
<tt> "Euler"</tt> uses the Euler product for computation.
The check omitted when using <tt> "fast"</tt> may be done later
by using <tt> OrderClassGroupCheck</tt>.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
We are going to compute the class group
of F = {\Bbb Q}(\sqrt[4]{-65}). 
<pre>

kash> o := OrderMaximal(Z,4,-65);;
kash> Time(true);
true
Time: 0 ms
kash> OrderClassGroup(o);
[ 128, [ 2, 4, 4, 4 ] ]
Time: 260 ms

</pre>

<p>

<p>
<hr>
<p>
<h3>
Example:
</h3>


The class group is of order 128 and is isomorphic to
C_2 \times C_4 \times C_4 \times C_4. \smallskip

In the example above, the time display was activated.
Compare the runtime difference when computing
the class group of F by taking 30 as a bound
for the prime ideals. Notice, that the Minkowski bound is 1274.
<pre>

kash> o := OrderMaximal(Z,4,-65);;
kash> OrderMinkowski(o);
1274
Time: 0 ms
kash> OrderClassGroup(o,30);
[ 128, [ 2, 4, 4, 4 ] ]
Time: 110 ms

</pre>

<p>


<hr>

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